Next, we discuss DM production through inflaton decays. The corresponding decay width averaged over inflaton oscillations is given in Eq. (REF ).
In this case, DM particles are efficiently produced only during the reheating period when the \(\phi \)  field is still abundant. After reheating, the inflaton number density dilutes rapidly, and the \(\phi \rightarrow XX\)  channel is effectively turned off. The predicted relic abundance of DM species produced by the inflaton decay can be calculated from (REF ). In Fig. REF  (center-panel) we present the parameter space in the \(\mathcal {C}_X^\phi -m_X\)  plane that reproduces the observed DM relic abundance, i.e. \(\Omega _X^{\rm obs} h^2 \!=\! 0.12\)  [1]},for three benchmark values of the inflaton-Higgs coupling \(g_{h \phi } =10^{-5}, 10^{-7}, 10^{-10}\) . One can see that for smaller value of \(g_{h \phi }\)  one needs weaker inflaton-DM coupling \(\mathcal {C}_{X}^{\phi }\)  to satisfy \(\Omega ^{\rm {obs}}h^2\)  constraint for a fixed DM mass. This behavior is related to the fact that for weaker inflaton-Higgs coupling the reheating lasts longer.
